Prove that matrix is invertible

**Julius** · February 14th 2010, 05:09 AM

A is a matrix, I is identity matrix and there exists such n that $A^n=4A^3+3A^2+2A+I$ and i have to prove that A is invertible. so i think we have to find such B that $AB=I$ , then $A(A^{n-1}-4A^2-3A-2)=I$ but obviously "-2" is bad here.. $AI=A$ so maybe $A(A^{n-1}-4A^2-3A-2)=AI(A^{n-1}-4A^2-3A-2)=A(A^{n-1}-4A^2-3A-2I)=I$ makes proof right?

**HallsofIvy** · February 14th 2010, 06:08 AM

Yes, that is correct. Well done!

(Although it would have made more sense to write 2A as (2I)A immediately.)

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